How do you find the area under the graph of #f(x)=x^3# on the interval #[-1,1]# ?

Question

Emilia Aguilar · Accepted Answer

Well, we need to be careful when you say "under the graph" since #f(x)=x^3# goes below the x-axis when #x<0#, but you meant the region between the graph and the x-axis, then the area of the region is 1/2. Since there are two regions: one from #x=-1# to #x=0# and the other from #x=0# to #x=1#, the area #A# can be found by
#A=int_{-1}^0(0-x^3)dx+int_0^1(x^3-0)dx#
by Power Rule,
#=[-x^4/4]_{-1}^0+[x^4/4]_0^1=1/4+1/4=1/2#

Charles Anctil · Answer

undefined To find the area under the graph of $f(x) = x^3$ on the interval $[-1, 1]$, you integrate the function from -1 to 1:

$$ \text{Area} = \int_{-1}^{1} x^3 \, dx $$

Integrate $x^3$ with respect to $x$ over the interval $[-1, 1]$. This yields the area under the curve between $x = -1$ and $x = 1$.

HIX Tutor · Answer

undefined To find the area under the graph of $ f(x) = x^3 $ from $-1$ to $1$, follow these steps:

1. **Understand the Graph**: The function $ f(x) = x^3 $ is symmetric about the origin. This means it has equal area above and below the x-axis.

2. **Set Up the Integral**: You need to find the integral of $ f(x) $ from $-1$ to $1$. This looks like this:
   $$
   \int_{-1}^{1} x^3 \, dx
   $$

3. **Calculate the Integral**: Find the antiderivative of $ x^3 $:
   $$
   \frac{x^4}{4}
   $$

4. **Evaluate the Antiderivative at the Bounds**:
   - Plug in $1$:
     $$
     \frac{1^4}{4} = \frac{1}{4}
     $$
   - Plug in $-1$:
     $$
     \frac{(-1)^4}{4} = \frac{1}{4}
     $$

5. **Subtract**: Now subtract the lower bound from the upper bound:
   $$
   \frac{1}{4} - \frac{1}{4} = 0
   $$

6. **Conclusion**: So, the area under the graph from $-1$ to $1$ is $0$. The areas above and below the x-axis cancel each other out!